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A well-known issue when testing for treatment-by-subgroup interaction is its low power, as clinical trials are generally powered for establishing efficacy claims for the overall population, and they are usually not adequately powered for detecting interaction (Alosh, Huque, & Koch [2015] Journal of Biopharmaceutical Statistics, 25, 1161-1178). Hence, it is necessary to develop an adaptive design to improve the efficiency of detecting heterogeneous treatment effects within subgroups. Considering Neyman allocation can maximize the power of usual Z-test (see p. 194 of the book edited by Rosenberger and Lachin), we propose a subgroup-adaptive randomization procedure aiming to achieve Neyman allocation in both predefined subgroups and overall study population in this paper. To verify whether the proposed randomization procedure works as intended, relevant theoretical results are derived and displayed . Numerical studies show that the proposed randomization procedure has obvious advantages in power of tests compared with complete randomization and Pocock and Simon's minimization method.

Background: Non-negative Matrix Factorization (NMF) has been extensively used in gene expression data. However, most NMF-based methods have single-layer structures, which may achieve poor performance for complex data. Deep learning, with its carefully designed hierarchical structure, has shown significant advantages in learning data features. Objective: In bioinformatics, on the one hand, to discover differentially expressed genes in gene expression data; on the other hand, to obtain higher sample clustering results. It can provide the reference value for the prevention and treatment of cancer. Method: In this paper, we apply a deep NMF method called Deep Semi-NMF on the integrated gene expression data. In each layer, the coefficient matrix is directly decomposed into the basic and coefficient matrix of the next layer. We apply this factorization model on The Cancer Genome Atlas (TCGA) genomic data. Results: The experimental results demonstrate the superiority of Deep Semi-NMF method in identifying differentially expressed genes and clustering samples. Conclusion: The Deep Semi-NMF model decomposes a matrix into multiple matrices and multiplies them to form a matrix. It can also improve the clustering performance of samples while digging out more accurate key genes for disease treatment.

The diagnosis of thyroid nodules at an early stage is a challenging task. Manual diagnosis of thyroid nodules is labor-intensive and time-consuming. Meanwhile, due to the difference of instruments and technical personnel, the original thyroid nodule ultrasound images collected are very different. In order to make better use of ultrasound image information of thyroid nodules, some image processing methods are indispensable. In this paper, we developed a method for automatic thyroid nodule classification based on image enhancement and deep neural networks. The selected image enhancement method is histogram equalization, and the neural networks have four-layer network nodes in our experiments. The dataset in this paper consists of thyroid nodule images of 508 patients. The data are divided into 80% training and 20% validation sets. A comparison result demonstrates that our method can achieve a better performance than other normal machine learning methods. The experimental results show that our method has achieved 0.901961 accuracy, 0.894737 precision, 1 recall, and 0.944444 F1-score. At the same time, we also considered the influence of network structure, activation function of network nodes, number of training iterations, and other factors on the classification results. The experimental results show that the optimal network structure is 2500-40-2-1, the optimal activation function is logistic function, and the best number of training iterations is 500.

In this paper, new refinements and improvements of Mitrinović–Cusa’s and related inequalities are presented. First, we give new polynomial bounds for sinc(x) and cos(x) functions using the interpolation and approximation method. Based on the obtained results of the above two functions, we establish new bounds for Mitrinović–Cusa’s, Wilker’s, Huygens’, Wu–Srivastava’s, and Neuman–Sándor’s inequalities. The analysis results show that our bounds are tighter than the previous methods.

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

A metamorphic linkage is capable of changing its motion branches and can be used as mechanisms for reconfigurable robots for various tasks. This paper presents two novel metamorphic linkages as the spherical-planar 6R metamorphic linkage and the Bennett-spherical 6R metamorphic linkage both of which have three various distinguished motion branches. Having established the close-loop equation of the spherical-planar 6R metamorphic linkage, the paper reveals the conditions of various motion branches and a set of transformations for switching motion branches. The paper further uses to reveal the inherent properties of this over-constrained metamorphic 6R linkage that is able to perform both spherical and planar motion with mobility one. Because of geometrical constraints at bifurcation points, the linkage is able to reconfigure to the deployed spherical motion branch, the planar motion branch and the folded spherical motion branch. The two spherical motion branches could be seen on both a large sphere that presents the deployed spherical motion and a small sphere that presents the folded spherical motion. This leads to the revelation of the novel Bennett-spherical 6R metamorphic linkage that has the transition from one deployed Bennett configuration branch to a spherical configuration branch and then to another folded Bennett configuration branch. Given the geometric parameters of both metamorphic linkages, it reveals that these linkages are special cases of Bricard line-symmetric 6R linkage.

The furcation point in a mechanism is a salient feature of reconfigurable mechanisms and change of a joint from an unconstrained condition to a geometrically constrained condition resulting in naturally link annexing with this geometrical constraint is a typical feature of metamorphic mechanisms. This paper presents a novel 6R metamorphic mechanism by inserting two revolute joints to a Bennett mechanism. From parameters of this novel 6R metamorphic mechanism, the source mechanism before changing is a special case of the Bricard 6R line-symmetric mechanism while the closure equation gives constraint conditions of joints. The geometrically constrained conditions result in variable motion branches of the mechanism. When all joints are unconstrained geometrically, two 6R motion branches can be obtained and when two joints are under geometric constrained, three 4R motion branches can be obtained. Further, three further motion branches with coaxial joints are obtained and motion branch transformation is illustrated with kinematic curves. For each of the motion branches, motion screws of this novel 6R metamorphic mechanism present corresponding geometry morphology and are analysed with screw algebra.

In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an 8π range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a 4π range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.

In this paper, new refinements and improvements of Jordan&rsquo, s and Kober&rsquo, s inequalities are presented. We give new polynomial bounds for the s i n c ( x ) and cos ( x ) functions based on the interpolation and approximation method. The results show that our bounds are tighter than the previous methods.

This paper studies contrastive divergence (CD) learning algorithm and proposes a new algorithm for training restricted Boltzmann machines (RBMs). We derive that CD is a biased estimator of the log-likelihood gradient method and make an analysis of the bias. Meanwhile, we propose a new learning algorithm called average contrastive divergence (ACD) for training RBMs. It is an improved CD algorithm, and it is different from the traditional CD algorithm. Finally, we obtain some experimental results. The results show that the new algorithm is a better approximation of the log-likelihood gradient method and outperforms the traditional CD algorithm.

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature, however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

Contrastive Divergence has become a common way to train Restricted Boltzmann Machines; however, its convergence has not been made clear yet. This paper studies the convergence of Contrastive Divergence algorithm. We relate Contrastive Divergence algorithm to gradient method with errors and derive convergence conditions of Contrastive Divergence algorithm using the convergence theorem of gradient method with errors. We give specific convergence conditions of Contrastive Divergence learning algorithm for Restricted Boltzmann Machines in which both visible units and hidden units can only take a finite number of values. Two new convergence conditions are obtained by specifying the learning rate. Finally, we give specific conditions that the step number of Gibbs sampling must be satisfied in order to guarantee the Contrastive Divergence algorithm convergence.

In this paper, the classical risk model perturbed by diffusion is considered in presence of a constant dividend barrier. We study the Gerber-Shiu discounted penalty function. Two integro-differential equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results of discounted penalty function are obtained.

In this paper, we consider the classical risk model with a linear dividend barrier. In this model, we study the Gerber-Shiu discounted penalty function. Two integro-differential equations for the Gerber-Shiu discounted penalty function are derived. The analytic results of discounted penalty function are obtained.

In this paper, we propose and analyze a two-grid variational multiscale method for the steady natural convection problem based on two local Gauss integrations technique. This method possesses the best algorithmic characteristics of both variational multiscale method and two-grid discretization. The main idea is to first solve the nonlinear steady natural convection problem on the coarse grid, then to use the coarse grid solution to fix the nonlinear terms, and to solve a linear problem on the fine grid. Stability and optimal error estimates of the discrete solutions in both one-grid and two-grid variational multiscale formulations are established. Finally, some numerical examples are presented to verify the method’s promise and testify the theoretical predictions.